Verify each identity. tan (A-B)=tan A-tan B/1+tan A tan B

The simplified trigonometric expression (cot²θ - csc²θ) / (tan²θ - sec²θ) simplifies to 0.

To simplify the trigonometric expression (cot²θ - csc²θ) / (tan²θ - sec²θ), we can use trigonometric identities.

Starting with the numerator: cot²θ - csc²θ

We know that cotθ = 1/tanθ and cscθ = 1/sinθ. Substituting these values, we get: (1/tanθ)² - (1/sinθ)²

Simplifying further: (1/tan²θ) - (1/sin²θ)

Using the identity tan²θ = 1 - cos²θ and sin²θ = 1 - cos²θ, we can rewrite the expression: (1/(1 - cos²θ)) - (1/(1 - cos²θ))

The denominators are the same, so we can combine the terms in the numerator: (1 - 1)/(1 - cos²θ)

Simplifying the numerator: 0/(1 - cos²θ) = 0

Now, let's move on to the denominator: tan²θ - sec²θ

Using the identity sec²θ = 1 + tan²θ, we can rewrite the expression:

tan²θ - (1 + tan²θ)

Combining like terms: -1

Therefore, the simplified expression is 0 / -1, which simplifies to 0. The simplified trigonometric expression (cot²θ - csc²θ) / (tan²θ - sec²θ) simplifies to 0.

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Approximately 2,052.19 gallons of water will leak from the faucet in one year.

To calculate the number of gallons of water that will leak from the faucet in one year, we need to convert the given measurements into consistent units.

Given:

1 fluid ounce every 30 seconds

First, let's convert the time from seconds to minutes since there are 60 seconds in a minute:

30 seconds ÷ 60 = 0.5 minutes

Next, let's calculate the number of minutes in one year:

60 minutes/hour * 24 hours/day * 365 days/year = 525,600 minutes/year

Now, let's calculate the total number of fluid ounces leaked in one year:

0.5 minutes * 525,600 minutes/year = 262,800 fluid ounces/year

Finally, let's convert the fluid ounces to gallons. There are 128 fluid ounces in a gallon:

262,800 fluid ounces ÷ 128 fluid ounces/gallon ≈ 2,052.19 gallons

Therefore, approximately 2,052.19 gallons of water will leak from the faucet in one year.

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The simplified expression is x(x + 8) / (x² - 2).

To simplify the expression (3x + x² + 5x) / (x² - 2), we can combine like terms in the numerator and denominator.

Starting with the numerator, we have 3x + x² + 5x. Combining the like terms, we get:

3x + x² + 5x = (3 + 5)x + x² = 8x + x²

Now let's simplify the denominator, which is x² - 2.

The given expression is:

(8x + x²) / (x² - 2)

Factoring the numerator as much as possible, we have:

8x + x² = x(x + 8)

Factoring the denominator, we have:

x² - 2

So the simplified expression becomes:

x(x + 8) / (x² - 2)

There are no specific restrictions mentioned for the variable x in the given expression. However, we should note that there might be restrictions depending on the context or any other related equations that are not explicitly provided.

In summary, the simplified expression is x(x + 8) / (x² - 2).

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The effective annual rate for 9% compounded daily is approximately 9.34%, for 9% compounded monthly is approximately 9.38%, and for 9% compounded yearly is 9%.

To calculate the effective annual rate, we use the formula: (1 + r/n)^n - 1, where r is the stated annual interest rate and n is the number of compounding periods per year.

For 9% compounded daily, the calculation is (1 + 0.09/365)^365 - 1, resulting in an effective annual rate of approximately 9.34%.

For 9% compounded monthly, the calculation is (1 + 0.09/12)^12 - 1, which gives an effective annual rate of approximately 9.38%. Finally, for 9% compounded yearly, there is no need for calculation as the stated interest rate and the effective annual rate are the same, which is 9%. The effective annual rate takes into account the compounding frequency and represents the total interest earned on an investment over a year. Higher compounding frequencies lead to slightly higher effective annual rates due to more frequent interest accumulation.

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R-squared value is approximately 0.7222, or 72.22%, indicating that approximately 72.22% of the total variation in the dependent variable. The residual for the first observation is 6.45.

The intercept between x and y occurs when the value of x is 0. However, since the given data only includes values for x greater than 0, we cannot directly determine the intercept. More information or additional data points are needed to determine the intercept.

To calculate the residual for the first observation, we can use the formula for the residual:

Residual = y1 - (intercept + slope * x1)

Given that the sample slope is 2.55, the sample intercept is 8.45, x1 is 22, and y1 is 70, we can substitute these values into the formula:

Residual = 70 - (8.45 + 2.55 * 22)

Now we can calculate the residual:

Residual = 70 - (8.45 + 55.1) = 70 - 63.55 = 6.45

Therefore, the residual for the first observation is 6.45.

The R-squared value, also known as the coefficient of determination, is a measure of how well the regression line fits the data. It represents the proportion of the total variation in the dependent variable (y) that can be explained by the independent variable (x).

The R-squared value can be calculated using the formula:

R-squared = Explained Sum of Squares / (Explained Sum of Squares + Unexplained Sum of Squares)

Given that the Explained Sum of Squares is 32500 and the Unexplained Sum of Squares is 12500, we can substitute these values into the formula:

R-squared = 32500 / (32500 + 12500) = 32500 / 45000 = 0.7222

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The inverse of the function [tex]\(y = \sqrt{x - 2} + 1\)[/tex] is [tex]\(f^{-1}(x) = x^2 - 2x + 3\)[/tex] , and it is indeed a function.

To find the inverse of the function [tex]\(y = \sqrt{x - 2} + 1\)[/tex], we can follow these steps:

Step 1: Swap x and y:

[tex]\(x = \sqrt{y - 2} + 1\)[/tex]

Step 2: Solve the equation for y:

[tex]\(x - 1 = \sqrt{y - 2}\)[/tex]

Step 3: Square both sides to eliminate the square root:

(x-1)² = y -2

Step 4: Simplify:

x² -2x + 1 = y- 2

Step 5: Add 2 to both sides:

x² - 2x + 3 = y

The inverse function of [tex]\(y = \sqrt{x - 2} + 1\)[/tex] is [tex]\(f^{-1}(x) = x^2 - 2x + 3\)[/tex] .

Now, to determine if the inverse is a function, we need to check if each input (x-value) of the original function corresponds to a unique output (y-value) in the inverse function.

In this case, both the original function and its inverse are functions.

The original function[tex]\(y = \sqrt{x - 2} + 1\)[/tex]  passes the horizontal line test, meaning each x-value maps to a unique y-value.

Similarly, the inverse function [tex]\(f^{-1}(x) = x^2 - 2x + 3\)[/tex] also passes the horizontal line test, indicating that it is a function.

Therefore, the inverse of the function [tex]\(y = \sqrt{x - 2} + 1\)[/tex] is [tex]\(f^{-1}(x) = x^2 - 2x + 3\)[/tex] , and it is indeed a function.

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To determine the rate constant (k) for a reaction based on the slope of a graph, we need to use the appropriate rate equation. The rate constant for this reaction is 0.0470.

In a first-order reaction, the rate equation is expressed as:

rate = k[A]

If the slope of the graph is given as 0.0470, we can equate it to the rate constant (k) in the first-order rate equation.

slope = k

Therefore, the rate constant for this reaction is 0.0470.

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In the repeated game G, each player has a total of 9 (pure) strategies.

In a repeated game, the strategy space for each player expands as they make decisions over multiple stages. Since the game G is repeated twice, we need to consider the number of possible strategy profiles in each stage and then multiply them to find the total number of strategies.

For Player 1, there are 3 possible actions in the first stage (a, b, c). In the second stage, given the knowledge of the strategy profile played in the first stage, Player 1 again has 3 possible actions. Therefore, the total number of strategies for Player 1 in the repeated game is 3 × 3 = 9.

Similarly, for Player 2, there are 4 possible actions in the first stage (W, X, Y, Z). In the second stage, Player 2 has 4 possible actions again. Hence, the total number of strategies for Player 2 in the repeated game is 4 × 4 = 16.

It's important to note that the total number of strategy profiles in the repeated game is the product of the number of strategies for each player. In this case, Player 1 has 9 strategies, and Player 2 has 16 strategies, resulting in a total of 9 × 16 = 144 strategy profiles in the repeated game.

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The exact value of sin(θ/2) is -√((√13 - 2) / 13)) / 2).To find the exact value of sin(θ/2), we'll need to use the given information about tanθ and the quadrant in which θ lies.

We know that tanθ = 3/2, which means the ratio of the opposite side to the adjacent side in a right triangle with angle θ is 3/2. Since 180° < θ < 270°, θ is in the third quadrant, where the tangent is positive.

In the third quadrant, the values of sine and cosine are negative. So, we can conclude that sinθ < 0 and cosθ < 0.

Now, let's use the half-angle formula for sine:

sin(θ/2) = ± √((1 - cosθ) / 2)

Since θ is in the third quadrant, sinθ is negative. Therefore, we can choose the negative sign in front of the square root in the formula:

sin(θ/2) = -√((1 - cosθ) / 2)

Substituting the given value of tanθ into the formula:

sin(θ/2) = -√((1 - cosθ) / 2)

         = -√((1 - (1 / √(1 + tan^2(θ)))) / 2)

         = -√((1 - (1 / √(1 + (3/2)^2)))) / 2)

         = -√((1 - (1 / √(1 + 9/4)))) / 2)

         = -√((1 - (1 / √(13/4)))) / 2)

         = -√((1 - (1 / (√13/2)))) / 2)

         = -√((1 - (2 / √13))) / 2)

         = -√((√13 - 2) / √13)) / 2)

         = -√((√13 - 2) / 13)) / 2)

Therefore, the exact value of sin(θ/2) is -√((√13 - 2) / 13)) / 2).

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By calculating the present value of the investment and determining the rate of return, we can assess the financial attractiveness and potential profitability of the investment opportunity.

(a) To calculate the present value of the investment, we need to discount the future cash flow of $5,000 back to the present using the appropriate discount rate of 8 percent. The formula for present value is given by PV = CF / (1 + r)^n, where PV is the present value, CF is the future cash flow, r is the discount rate, and n is the number of periods. By substituting the given values into the formula, we can calculate the present value.

(b) In this part, we are provided with the information that the investment is selling for $1,061 and will yield $5,000 in 33 years. We need to determine the rate of return investors earn on this investment. The rate of return, also known as the yield or internal rate of return (IRR), is the rate at which the investment grows over time. By using the formula for rate of return and rearranging it to solve for r, we can determine the rate of return when the investment is purchased for $1,061 and yields $5,000 in 33 years.

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We can conjecture that the equation [tex]1 + tan^2\theta = sec^2\theta[/tex] is a trigonometric identity and can be considered an alternative form or extension of the Pythagorean Identity, involving the tangent and secant functions.

Based on the given equation [tex]1 + tan^2\theta = sec^2\theta[/tex] , we can make a conjecture about this equation using our knowledge of trigonometric identities.

One commonly known identity is the Pythagorean Identity, which states that [tex]sin^2\theta + cos^2\theta = 1[/tex].

By rearranging the given equation, we can see a similarity to the Pythagorean Identity:

[tex]1 + tan^2\theta = sec^2\theta\\tan^2\theta + 1 = sec^2\theta[/tex]

Comparing this to the Pythagorean Identity, we can see that [tex]tan^2\theta[/tex] is equivalent to [tex]sin^2\theta[/tex] and 1 is equivalent to [tex]cos^2\theta[/tex].

Therefore, based on this observation, we can conjecture that the equation [tex]1 + tan^2\theta = sec^2\theta[/tex] is a trigonometric identity and can be considered as an alternative form or extension of the Pythagorean Identity, involving the tangent and secant functions.

However, it's important to note that a conjecture is a statement based on observation or reasoning and should be proven to be true using rigorous mathematical methods before it can be considered a valid identity.

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The probability of having a license would be affected by the probability of being an adult.

Given that we need to determine if the event of having a license and the event of being an adult independent event,

The event of having a license and the event of being an adult may or may not be independent events, depending on the specific context and circumstances.

To determine if two events are independent, we need to check if the occurrence or non-occurrence of one event affects the probability of the other event.

For example, let's consider a scenario where having a license refers to possessing a valid driver's license, and being an adult means being at least 18 years old. In many jurisdictions, obtaining a driver's license is typically age-dependent, where individuals can only acquire a license once they reach a certain age (e.g., 16 or 18 years old).

In this case, the events of having a license and being an adult are likely not independent. Being an adult is a prerequisite for obtaining a license in this context.

Therefore, the probability of having a license would be affected by the probability of being an adult.

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a. One possible nonzero linear function f(x) that has a zero at x=3 is f(x) = 2x - 6.

A linear function is represented by the equation f(x) = mx + b, where m is the slope of the line and b is the y-intercept. In this case, we want the function to have a zero at x=3, which means that when x=3, f(x) should equal zero.

Let's substitute x=3 into the function f(x) = 2x - 6:

f(3) = 2(3) - 6

f(3) = 6 - 6

f(3) = 0

Since f(3) equals zero, the function satisfies the condition of having a zero at x=3.

To find this function, we can start with the general form of a linear function f(x) = mx + b and then adjust the values of m and b to meet the given condition. By setting the slope m as 2 (which determines the steepness of the line) and the y-intercept b as -6 (which determines where the line intersects the y-axis), we obtain a linear function that has a zero at x=3.

Overall, the nonzero linear function f(x) = 2x - 6 has a zero at x=3 and can be represented by the equation f(x) = 2x - 6.

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Unfortunately, the question seems to be incomplete as it mentions "Consider the following two person game. (5)" without providing any details about the game, its payoff matrix, or any specific instructions. Without this information, it is not possible to draw the graphs or provide an analysis of Nash equilibria.

In general, to find the Nash equilibria of a game, one must analyze the players' strategies and payoffs. The Nash equilibrium is a situation in which no player has an incentive to unilaterally deviate from their chosen strategy. It is often identified by looking for stable points where the players' best responses intersect.

A common approach to finding Nash equilibria is through the analysis of strategic dominance, where strategies that are strictly dominated by others can be eliminated from consideration. This process reduces the strategic space and narrows down the possible equilibria.

However, without the specific game and its details, it is not possible to provide a diagram or analyze the Nash equilibria. It would be helpful to provide more information or the specific game setup to proceed with a meaningful analysis.

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without the actual data for the right-hand side variables, it is not possible to provide a specific predicted value. The solution would require the actual values of X variables, their median, and the estimated coefficients from the regression model.

To estimate the population regression model, we need to have data for the right-hand side variables (not provided in the question). The regression model specified in the question is incomplete without actual data, as we need the values of the independent variables (X) to make predictions.

However, if we had the data, we could estimate the regression coefficients using statistical methods such as ordinary least squares (OLS). Once we have the estimated coefficients, we can use them to predict the dependent variable (Y) at specific values of the independent variables.

To predict at the sample median of the right-hand side variables, we would substitute the median values into the regression equation. This would involve replacing the X variables in the equation with their respective median values. By calculating the predicted value using the estimated coefficients and the median values, we would obtain the predicted value at the sample median.

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The mean of the set of values {15, 17, 19, 20, 14, 23, 12} is 17. The standard deviation is approximately 3.97.

To find the mean of a set of values, we sum all the values and divide by the total number of values. In this case, the sum is 120 and there are 7 values, so the mean is 120/7 = 17.

To find the standard deviation, we first calculate the deviations of each value from the mean by subtracting the mean from each value. Then, we square each deviation, sum them up, divide by the total number of values, and take the square root of the result.

The deviations from the mean are {-2, 0, 2, 3, -3, 6, -5}. Squaring these deviations gives {4, 0, 4, 9, 9, 36, 25}. Summing them up gives 87.

Dividing by the total number of values, which is 7, gives 12.43. Taking the square root gives approximately 3.97, which is the standard deviation of the set of values.

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Answer:

A and B

Step-by-step explanation:

As for A, we see the segment of Y to X, and X continues onward as a line. As for B, we have the angle of ZXY, which is the angle that we can see right now. The center variable is the middle of the angle. This angle could also be called <YXZ.

No, the equations "a = 3sin 22°/sin 45°" and "a = 3sin(22/45)°" are not the same.

In the equation "a = 3sin 22°/sin 45°," the sine function is being applied individually to the angles of 22° and 45°, and the result is divided. This means that the sine of 22° is divided by the sine of 45°.

On the other hand, in the equation "a = 3sin(22/45)°," the angle 22/45 is being evaluated first, and then the sine function is applied to this value. So, the sine of (22/45)° is calculated.

The key difference between the two expressions is the order of operations. In the first equation, the division is performed after evaluating the sine functions for each angle separately. In the second equation, the division is performed after evaluating the sine of the angle obtained by dividing 22 by 45.

Therefore, these two equations represent different calculations and will yield different results. To ensure accurate calculations, it's important to apply the correct order of operations when using trigonometric functions.

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Step-by-step explanation:

92  is worth  70%

92 * .72    + 100 * .30 = 94.4 score

 If   "A"   is 93 or above....it looks like you got one !

       Therefore, Your Grade is Now:  94.4%

       Calculate the percentage of the summative grade:

       0.30  *  100   =  30

        1  -  0.30  = 0.70

        0.70  *  92  =  64.4

        64.4  +  30  =  94.4

        Therefore, Your Grade is Now:  94.4%

I hope this helps!

The factored form of the polynomial x³ - 36x is x(x + 6)(x - 6).


To factor the polynomial x³ - 36x, we look for common factors and apply factoring techniques.

The common factor in this polynomial is x. By factoring out x, we get x(x² - 36).

Next, we have a difference of squares expression x² - 36. This can be factored as (x + 6)(x - 6), where we use the pattern (a² - b²) = (a + b)(a - b).

Combining these factors, we obtain the factored form of the polynomial as x(x + 6)(x - 6).

To check the factored form, we can multiply the factors together and verify if it equals the original polynomial:

x(x + 6)(x - 6) = x(x² - 6x + 6x - 36) = x(x² - 36) = x³ - 36x.

As the result matches the original polynomial x³ - 36x, we can confirm that the factored form x(x + 6)(x - 6) is correct.

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The solution to the inequality is z ≈ 5.5

Given the expression :

We can solve for x thus:

-2z + 15 ≈ 4

subtract 15 from both sides

-2z ≈ 4 - 15

-2z ≈ -11

divide both sides by -2 to isolate z

z ≈ 5.5

Therefore, the value of z in the expression is 5.5

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In rectangles all the angles are of 90 degrees but in parallelograms it is not mandatory .

Here,

The opposite sides of parallelograms and rectangles are equal and parallel.

So on the basis of sides the parallelograms are same as rectangles but the difference is created by the angles of rectangle and parallelogram .

In rectangle all the angles are of 90 degrees but in the case of parallelogram all the angles need not to be 90 degrees always .

Thus all rectangles are parallelograms but all parallelograms are not rectangles.

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All rectangles are parallelograms because a rectangle has opposites sides parallel and equal, but all parallelograms are not rectangles because  a rectangle also has four right angles, which is not a property of  parallelograms.

A parallelogram is defined as a quadrilateral with opposite sides that are parallel. Since a rectangle has opposite sides that are parallel, it satisfies the definition of a parallelogram. Additionally, a rectangle also has four right angles, making it a special case of a parallelogram.

On the other hand, not all parallelograms are rectangles because a parallelogram can have various angles that are not necessarily right angles. While a rectangle has four right angles, a parallelogram can have any angle measurements as long as its opposite sides are parallel. This means that a parallelogram can have acute angles, obtuse angles, or a combination of both. Therefore, a parallelogram that does not have four right angles does not meet the specific criteria to be considered a rectangle.

All rectangles are parallelograms, but all parallelograms are not rectangles

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To calculate the average speed for the entire trip, we need to know the total distance traveled.

To calculate the average speed for the entire trip, we need to know the total distance traveled and the total time taken. Since the individual drove back home using the same path taken to the university, the total distance covered will be twice the distance from home to the university.

Let's denote the distance from home to the university as "d" kilometers. Therefore, the total distance traveled is 2d kilometers.

Given that the individual arrives back home 7.9 hours after initially leaving, we need to find the total time taken for the round trip. The total time consists of the time taken from home to the university and the time taken from the university back home.

Let's denote the average speed for the entire trip as "s" kilometers per hour.

We can use the formula: speed = distance / time

1. Time taken from home to university:

  Distance: d kilometers

  Time: t₁ hours (unknown)

  Speed₁ = d / t₁

2. Time taken from university back home:

  Distance: d kilometers

  Time: t₂ hours (unknown)

  Speed₂ = d / t₂

Since the individual arrives back home after 7.9 hours, the total time taken is the sum of t₁ and t₂:

t₁ + t₂ = 7.9

We want to find the average speed for the entire trip, which is the total distance (2d) divided by the total time (t₁ + t₂):

Average speed = Total distance / Total time

             = 2d / (t₁ + t₂)

To calculate the average speed, we need to find the values of t₁ and t₂. We can do this by solving the equation t₁ + t₂ = 7.9 using the given information.

Once we have the values of t₁ and t₂, we can substitute them into the average speed formula to calculate the average speed for the entire trip.

In summary, to determine the average speed for the entire trip, we need to find the values of t₁ and t₂ by solving the equation t₁ + t₂ = 7.9. Once we have these values, we can calculate the average speed using the formula 2d / (t₁ + t₂), where "d" represents the distance from home to the university.

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In this study, one individual refers to a single third-grade student from the total population of third graders in the school.

The student collected information on the number of siblings for each member of her class, which consists of 22 students. However, to consider the entire population, we need to take into account all the third-grade classes in the school. Since the school has five different third-grade classes, the population of interest comprises all the third graders across these five classes.

Each student in the population is considered an individual for the study. Therefore, one individual in this context refers to any random third-grade student from the school, regardless of the specific class they belong to. To conduct a comprehensive study and obtain accurate information about the number of siblings among third graders in the school, it would be necessary to collect data from a representative sample across all the third-grade classes.

By doing so, researchers can make inferences and draw conclusions about the entire population of third graders in the school based on the collected data.

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The solutions of the equation 81x³-192=0 are x=-2, x=2, and x=0. We can solve the equation by factoring 81x³-192. The first step is to factor out a 81 from the left side of the equation.

This gives us 81x³-192 = 81(x³-2). The next step is to factor x³-2. We can do this by using the difference of cubes factorization. This gives us x³-2 = (x-2)(x²+2x+1). Finally, we can equate each factor to 0 and solve for x. This gives us x=2, x=-2, and x=-1/2.

81x³-192 = 81(x³-2)

= 81(x-2)(x²+2x+1)

(x-2)(x²+2x+1) = 0

x-2 = 0 or x²+2x+1 = 0

x = 2 or x = -2 or x = -1/2

The first step is to factor out a 81 from the left side of the equation. This gives us 81x³-192 = 81(x³-2).

The second step is to factor x³-2 using the difference of cubes factorization. This gives us x³-2 = (x-2)(x²+2x+1).

The third step is to equate each factor to 0 and solve for x. This gives us x=2, x=-2, and x=-1/2.

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The measure of the inscribed angle y in the circle is 60 degrees.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

An inscribed angle is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the diagram:

Inscribed angle y =?

Intercepted arc of angle y = 120 degrees

Plug the given value into the above formula and solve for the Inscribed angle y:

Inscribed angle = 1/2 × intercepted arc.

Inscribed angle y = 1/2 × 120°

Inscribed angle y = 60°

Therefore, angle y measures 60 degrees.

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After serving the 3 glasses, approximately 12% of the bottle will remain.

We need to determine how much liquid is used to fill the glasses and then subtract that amount from the total volume of the bottle.

Each glass requires 220ml of lemonade, and since there are 3 glasses, the total amount used to fill the glasses is 220ml/glass × 3 glasses = 660ml.

To find the remaining amount of lemonade in the bottle, we subtract the amount used (660ml) from the total volume of the bottle (750ml):

Remaining amount = Total volume - Amount used

Remaining amount = 750ml - 660ml

Remaining amount = 90ml

Therefore, after serving the 3 glasses, there will be 90ml of lemonade remaining in the bottle.

To calculate the percentage, we divide the remaining amount by the total volume of the bottle and multiply by 100:

Percentage remaining = (Remaining amount / Total volume)×100

Percentage remaining = (90ml / 750ml) × 100

Percentage remaining = 0.12 × 100

Percentage remaining = 12%

So, after serving the 3 glasses, approximately 12% of the bottle will remain.

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A lemonade bottle has 750ml. If you need to fill 3 glasses of 220ml each, what percentage of the bottle remains after serving the 3 glasses?

In terms of the sine, we can say that cos 16 can be written as:

Option B: Sine 74°

We know that all the trigonometric identities are based on the six trigonometric ratios. They are identified as sine, cosine, tangent, cosecant, secant, and cotangent

For complementary angles; or acute angles, we know that:

Sin x = Cos y,

This means that:

x + y = 90

Thus, if x = 16 then it means that:

y = 90 - 16 = 74

Hence;

Cos 16 = (sine 90-6)

Cos 16 = Sine 74

We conclude that in terms of sine; cos 16 is equal to Sine 74°

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Neither of the base angles of an isosceles triangle can be a right angle.

An isosceles triangle has two sides of equal length. The base angles are the angles at the base of the triangle, opposite the two equal sides. A right angle is an angle that measures 90 degrees.

In order for an angle to be a right angle, it must be formed by two perpendicular lines. Perpendicular lines are lines that intersect at a right angle. In an isosceles triangle, the base angles are opposite the equal sides. If one of the base angles were a right angle, then the two equal sides would be perpendicular. However, this is not possible, as perpendicular lines can only intersect once. Therefore, neither of the base angles of an isosceles triangle can be a right angle.

Here is an illustration of an isosceles triangle with two right angles:

```

[asy]

unitsize(0.5 cm);

pair A, B, C;

A = (0,0);

B = (2,0);

C = (1,sqrt(3));

draw(A--B--C--A);

draw(rightanglemark(A,B,C,20));

draw(rightanglemark(A,C,B,20));

label("$A$", A, SW);

label("$B$", B, SE);

label("$C$", C, NE);

[/asy]

```

As you can see, the two base angles of this triangle are both right angles. However, this is not a valid isosceles triangle, as the two equal sides are not perpendicular.

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The polynomial that passes through the points is y = x⁴ + 2x³ + 3x² - x + 1

From the question, we have the following parameters that can be used in our computation:

(-3, 58), (-2, 15), (1, 6), (2, 43), (5, 946)

Given that there are 5 points

This means that the degree of the polynomial is 4

And it can be represented as

y = ax⁴ + bx³ + cx² + dx + e

Using the points, we have

a(-3)⁴ + b(-3)³ + c(-3)² + d(-3) + e = 58

a(-2)⁴ + b(-2)³ + c(-2)² + d(-2) + e = 15

a(1)⁴ + b(1)³ + c(1)² + d(1) + e = 6

a(2)⁴ + b(2)³ + c(2)² + d(2) + e = 43

a(5)⁴ + b(5)³ + c(5)² + d(5) + e = 946

So, we have

81a - 27b + 9c - 3d + e = 58

16a - 8b + 4c - 2d + e = 15

a + b + c + d + e = 6

16a + 8b + 4c + 2d + e = 43

625a + 125b + 25c + 5d + e = 946

When evaluated, we have

a = 1 and b = 2 and c = 3 and d = -1 and e = 1

So, we have

y = x⁴ + 2x³ + 3x² - x + 1

Hence, the polynomial is y = x⁴ + 2x³ + 3x² - x + 1

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